Heat oven to 425º F with a rack in the lower middle position. Spray a sheet pan with cooking spray, then cover with parchment paper. (Alternately use a Silpat)
Combine ¾ cup water, milk, butter and salt in a 2 quart heavy saucepan. Heat on medium-low, and bring the mixture to a boil. (The butter should melt before it boils.)
Meanwhile whisk the flour for about 30 seconds to aerate and break up any clumps. In a 2 cup liquid measure combine the 3 eggs and 1 egg white and mix to combine. In a small bowl whisk together yolk and 2 tablespoons water (eggwash.)
Once the water has just started to boil, remove it from the heat and dump all of the flour in at once. Stir the mixture with a heat proof spatula until the flour is mostly incorporated. Return to the heat, and use the spatula to smear the mixture across the bottom of the pan continuously for 3 to 4 minutes. A thin film of flour will have build up on the bottom of the pan.
Transfer the dough to the work bowl of a food processor fitted with the blade attachment. Process for 15 seconds to cool slightly. With the processor running add the eggs through the feed tube in a stream over about 15 seconds. Stop the processor and scrape down the sides of the bowl. Process an additional 30 to 40 seconds until a smooth paste has formed.
Transfer the paste to a gallon plastic bag, cut a ½- to ¾-inch diameter opening in one corner. Pipe the dough into a 2 ½- to 2 ¾-inch circles about 1-inch high, making sure to leave about 2 inches between puffs. Brush the puffs with the eggwash using the pastry brush to smooth the tops of the puffs in addition to coating the puffs. Take care not to get the eggwash onto the baking sheet as it will burn.
Bake 10 minutes, then reduce heat to 375º F and bake an additional 17 to 20 minutes. The puffs should be a deep golden brown. Turn oven off. Use a paring knife to cut a slit in the side of each puff (where you plan on cutting them in half.) Place the tray of puffs back in the oven, and use a wooden spoon to crack the door open. Allow the puffs to dry in the oven for 60 minutes. Remove from oven and cool on rack before splitting. Can be stored in airtight container at room temperature for a day or two - recrisp for 3 minutes in a 350º oven before serving.
To serve cut puffs in half, fill with about 1/3rd cup of the whipped cream filling, and dust with confectioner's sugar.
Bourbon Chocolate Whipped Cream (adapted from Rose Levy Beranbaum)
1 cup heavy whipping cream
2 tablespoons Dutched cocoa powder
2 tablespoons, plus 2 teaspoons granulated sugar
1 tablespoon bourbon (I used Knob Creek)
Combine cream, cocoa, and sugar in a medium bowl and whisk to combine. Cover and refrigerate at least an hour to allow the cocoa to dissolve. Whisk by hand or beat with a hand-mixer until stiff peaks form. Whisk in the bourbon. Use to fill cream puffs.
Cook's note: if you don't want to use bourbon, whisk in a teaspoon of vanilla extract instead to round out the chocolate flavor. Source